Endergonic reactions require an input of energy, so yes, they require ATP. ATP catabolises to form ADP and Pi, releasing energy as it does. The energy released is then used in the endergonic reaction.
An
example of an endergonic reaction is the synthesis of ATP from ADP and Pi. Remember that ATP has three phosphates while ADP only had two. How is ADP, with its two phosphates, going to bond with Pi? It needs energy. Energy has to be released from other catabolic reactions to make the Pi bond with ADP to form ATP.
Yes, ATP is catabolised to form ADP and Pi in order to release energy in an exergonic reaction. Cells use energy in the form of ATP because this phosphate bond can easily be broken. Think of ATP as a muesli bar or something - when you snap it apart to eat it you release energy in the form of sound, and to make the muesli bar you need to cook it or something. When a cell needs energy for an endergonic reaction, ATP can diffuse through, catabolise, and "release" energy which is taken up by the reactions.
Exergonic reactions do release energy but it doesn't just have to be in ATP. It can also release energy in the form of heat. Exergonic reactions just give enough energy for Pi and ADP to bond to form ATP.
If you need any more clarification watch this:
http://www.youtube.com/watch?v=AhuqXwvFv2E